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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Nonlocal invariants in index theory

Author: Steven Rosenberg
Journal: Bull. Amer. Math. Soc. 34 (1997), 423-433
MSC (1991): Primary 58G25; Secondary 58G10, 58G25, 58G26
MathSciNet review: 1458426
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Abstract: In its original form, the Atiyah-Singer Index Theorem equates two global quantities of a closed manifold, one analytic (the index of an elliptic operator) and one topological (a characteristic number). Because it relates invariants from different branches of mathematics, the Index Theorem has many applications and extensions to differential geometry, K-theory, mathematical physics, and other fields. This report focuses on advances in geometric aspects of index theory. For operators naturally associated to a Riemannian metric on a closed manifold, the topological side of the Index Theorem can often be expressed as the integral of local (i.e. pointwise) curvature expression. We will first discuss these local refinements in §1, which arise naturally in heat equation proofs of the Index Theorem. In §§2,3, we discuss further developments in index theory which lead to spectral invariants, the eta invariant and the determinant of an elliptic operator, that are definitely nonlocal. Finally, in §4 we point out some recent connections among these nonlocal invariants and classical index theory.

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Additional Information

Steven Rosenberg
Affiliation: Department of Mathematics, Boston University, Boston, Massachusetts 02215

Additional Notes: Partially supported by the NSF
Article copyright: © Copyright 1997 American Mathematical Society